The transition from the sup-expression to \(r^\nu (x)\)  is achieved by recognizing that the former is the conjugate of the function \(u\mapsto {\raisebox {1pt}{\frac{1}{2}}}\theta ^\nu \Vert u\Vert _2^2 + \iota _{\Delta }(p^\nu + u)\) , evaluated at a certain point. This in turn is expressed by the conjugates of \({\raisebox {1pt}{\frac{1}{2}}}\theta ^\nu \Vert \cdot \Vert _2^2\)  and \(\iota _{\Delta }\) ; see [1]} as well as Equation 11(3) and Theorem 11.23(a) in [2]}.
